Aristotle did say in Politics IV that appointing public office by lottery, drawing from the real public, is more democratic (power to the people) than elections, which is an oligarchic exercise.
I've always held the opinion that elected officials should have to use public health care, send their kids to public schools, and use public transport.
This would ensure that they would have to maintain these institutions and be able to face their constituents on the daily.
-- Gustavo Petro
Obviously OP is not saying it was unilaterally better and the idea being commented on is just public lottery vs election. You can lift out that idea and apply our current understanding of “public” and the point of OP still stands.
By excluding over 3/4 of the population, we would hardly get what we think of as democracy.
With those exclusions, the power of being a citizen would increase dramatically and the lottery would make it unlikely bands of eligible citizens would unite to expand the electorate but instead would seek to make membership more stringent, like DPRK's haeksim.
Choice by lottery would also fail to exclude men who have no desire or capacity to do whatever job it is they pulled. I do not know what they did in ancient Greece but in modern times that would lead to a bureaucratic class with the leader being only a figurehead at their mercy--and at times that would be very good. But the bureaucrats class would not be the voice of the people, would not be democratic.
I felt this was obvious from the exclusions and so didn't see the need to elaborate.
More like consolidating and cementing. It had already been built out